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Is growth linear, not exponential?

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Re: Is growth linear, not exponential?

#21

Post by ajuc »

jasoncrawford wrote: post #4
apolloartemis wrote: post #2

Can’t you model almost anything as a piecewise linear function? I don’t know if this claim is saying much of substance.

Yes, but if your data is actually exponential, the linear segments are not going to be better approximations than an exponential curve. That's what's going on here

For any sampled data you'll get guaranteed 100% fit by making it pieceways constant with N fragments where N=number of data points. It says nothing about the function you sampled, it's just a way to cheat by overfitting.

Re: Is growth linear, not exponential?

#23

Post by leetcrew »

heavenlyblue wrote: post #15
aidenn0 wrote: post #3

Is this new? I remember reading an article a long time ago (maybe 15 years?) discussing how growth in aircraft speed was overall exponential, but linear for each new technology that was introduced.

How is aircraft limited to the speed of sound still?

they're not? it just turns out that the most efficient cruising airspeed for passenger jets is in the upper end of the subsonic range. most people can't afford supersonic travel, and the ones that can overwhelming prefer to spend the extra money on a more comfortable subsonic flight.

interestingly, extremely high speeds turn out not to be that useful for military aircraft either. the fastest production fighter jet of all time, the mig-25, was introduced in 1970. the f-22 has a considerably lower top speed, and it's not because they lacked the funding to make it go faster.

Re: Is growth linear, not exponential?

#25

Post by jasoncrawford »

Enginerrrd wrote: post #19
jasoncrawford wrote: post #4

Earlier quoted context omitted.

Yes, but if your data is actually exponential, the linear segments are not going to be better approximations than an exponential curve. That's what's going on here

I'm not sure that's true in general, nor even frequently. In fact, I'd say it's provably false in general. The big issue is that you get MANY more curve-fitting parameters to play with if you use a piece-wise linear model vs. an exponential model. (You get to choose HOW MANY breaks to make, what the slope is for each section, and WHERE to make the breaks.) So... Let's say you created some synthetic data using an unde…

Good point.

The paper does cross-validate the models, and I am told that cross-validation properly penalizes overfitting with too many parameters… but I don't understand the statistics well enough here.

Re: Is growth linear, not exponential?

#26

Post by jasoncrawford »

pishpash wrote: post #17
dash2 wrote: post #6

Earlier quoted context omitted.

It would be a change to macro models of economic growth, if accepted. The default is that TFP grows exponentially. There's also a useful comment by Marginal Revolution here ( https://marginalrevolution.com/marginalrevolution/2022/04/ad... ) where he points out that TFP itself is really just a residual, and perhaps not a very well-defined concept. That is, if you regress Y = AK^beta L^(1-beta) with K being the amount…

Why is TFP assumed exponential by default? What's the theory behind that? A time lag in the transfer between exponential input growth (e.g. population) and exponential output growth?

It's not always assumed to be exponential. It was assumed exponential by Paul Romer because long-term economic growth is pretty exponential. Much subsequent work has followed that model, but not all of it.

Re: Is growth linear, not exponential?

#27

Post by robotresearcher »

heavenlyblue wrote: post #15
aidenn0 wrote: post #3

Is this new? I remember reading an article a long time ago (maybe 15 years?) discussing how growth in aircraft speed was overall exponential, but linear for each new technology that was introduced.

How is aircraft limited to the speed of sound still?

Adding detail to leetcrew's good reply: the energy of an aircraft in flight is proportional to the square of its velocity. Given steady progress in plane technology you'd expect practical top speed to be limited by cost of energy before too long. That happened more than thirty years ago.

Re: Is growth linear, not exponential?

#28

Post by 21723 »

It's almost certainly exponential, but the rate of growth depends on a number of dynamic factors. Corrupt elites often shut growth down. It happened in China and Japan several times in the second millennium, and it's happening in the US in the third one; we've backslid since about 1970.

Re: Is growth linear, not exponential?

#29

Post by lumost »

LeegleechN wrote: post #16
Fomite wrote: post #12

Earlier quoted context omitted.

The dangerous bit is that an exponential curve will also be a fairly good fit for a logistic function that's not yet fully observed.

Every apparent exponential in the real universe must actually be a logistic or some other bounded curve.

Except for the size of the Hubble volume /s

Cosmic inflation should guarantee an exponentially growing observable universe.

Re: Is growth linear, not exponential?

#30

Post by lumost »

leetcrew wrote: post #23
heavenlyblue wrote: post #15

Earlier quoted context omitted.

How is aircraft limited to the speed of sound still?

they're not? it just turns out that the most efficient cruising airspeed for passenger jets is in the upper end of the subsonic range. most people can't afford supersonic travel, and the ones that can overwhelming prefer to spend the extra money on a more comfortable subsonic flight. interestingly, extremely high speeds turn out not to be that useful for military aircraft either. the fastest production fighter jet of…

This is largely just military doctrine. Stealth was determined to be a better bet than speed against SAMs and other jets. With improved signal processing techniques and better standoff weapon, speed may be the better bet.
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